Projection of spin

Question

I have a naive and stupid question. In university we learn that particles, i.e. electrons have spin $1/2$. This spin can have 2 projections on some axis that we chose with the values $1/2$ and $-1/2$. Also we learn in the beginning of QM, that if particle can be in some states, it can also be in a linear combination of states.

So my first question is: Can projection on axis for spin-$1/2$ be any number between $1/2$ and $-1/2$. My second question: If particle has momentum in $z$ direction, can spin projection of spin be $0$ on z-axis?

Answer 1

A partcile can be in a superposition of two spin states (in respect to a chosen quantization axis), $$|\psi\rangle = c_1|\uparrow\rangle + c_2|\downarrow\rangle.$$ In every measurement we obtain either $+1/2$ or $-1/2$. However the mean of these measurements (which approaches the expectation value) can be anywhere in between.

In non-relativistic case the momentum is not coupled to spin, that is momentum can be zero, but the spin will still behave as described above. In the relativistic case (or if we include spin-orbit coupling, which is usually due to relativistic effects) the spin and the momentum are not independent, so one usually choses the direction of momentum as the quantization axis.